Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^2$ boundedness

Abstract

We prove the $L^2$ boundedness of the directional Hilbert transform in the plane relative to measurable vector fields which are constant on suitable Lipschitz curves. One novelty of our proof lies in the definition of the adapted Littlewood–Paley projection (see Definition 3.3). The other novelty is that we will use Jones’ beta numbers to control certain commutator in the critical Lipschitz regularity (see Lemma 5.5).